Morley’s Miracle (quoting from ‘Cut the Knot’) can be stated as follows:
The three points of intersection of the adjacent trisectors of the angles of any triangle form an equilateral triangle.
Let’s call the process referenced in this theorem of Morley’s the ‘Morley process’.
I would like to record here my conjecture, based on numerical evidence that I obtained, regarding what happens when the obvious modification is made to the Morley process, namely, that the trisections of the sides, instead of trisections of the angles, are considered.
Conjecture: If you start with any triangle T0 and from each vertex draw two line segments to the opposite side such that the two line segments trisect the opposite side (rather than trisecting the angle at that vertex), then, as a result of this modified Morley process, the pairwise intersections of adjacent side-trisectors determine a triangle T1 such that if you reflect T1 through the horizontal axis, obtaining triangle T2, and then you reflect T2 through the vertical axis, obtaining triangle T3, and then multiply the side-lengths of T3 by 5, obtaining triangle T4, then T4 can be made to coincide with T0 by means of at most one rotation and one translation. In a word, it appears that the smaller triangle is similar to, and exactly one fifth the size of, the original triangle. We might call the smaller triangle the ‘trinity triangle’ of the original triangle.
It is an easy matter to prove the conjecture for the case of an equilateral triangle, by placing it in the coordinate plane and obtaining the two equations for the two lines determining one of the vertices of the corresponding trinity triangle, solving them simultaneously, and combining knowledge of the coordinates of that vertex with the known proportionality information about the triangle, and I suspect this same technique would work in general.
keywords: Mathematics, Geometry, Morley’s Miracle, trisection